Abstract: | Let n 3 and
be positive integers, f :Sn→Sn be a C0-mapping, and
denote the standard embedding. As an application of the Pontryagin–Thom construction in the special case of the two-point configuration space, we construct complete algebraic obstructions O(f) and
to discrete and isotopic realizability (realizability as an embedding) of the mapping J f. The obstructions are described in terms of stable (equivariant) homotopy groups of neighborhoods of the singular set Σ(f)={(x,y) Sn×Sn f(x)=f(y), x≠y}.A standard method of solving problems in differential topology is to translate them into homotopy theory by means of bordism theory and Pontryagin–Thom construction. By this method we give a generalization of the van-Kampen–Skopenkov obstruction to discrete realizability of f and the van-Kampen–Melikhov obstruction to isotopic realizability of f. The latter are complete only in the case d=0 and are the images of our obstructions under a Hurewicz homomorphism. We consider several examples of computation of the obstructions. |